It is easy to see that the g.f. for indecomposable tilings (pavings) i.e. those that cannot be split vertically into smaller tilings, is g=2x+3x^2+2x^3+2x^4+2x^5+...=x(2+x-x^2)/(1-x); then G.f.=1/(1-g)=(1-x)/(1-3x-x^2+x^3). - Emeric Deutsch, Oct 16 2006